Designing electrodes in a highly ordered structure simultaneously with appropriate orientation, outstanding mechanical robustness, and high electrical conductivity to achieve excellent electrochemical performance remains a daunting challenge. Inspired by the phenomenon in nature that leaves significantly increase exposed tree surface area to absorb carbon dioxide (like ions) from the environments (like electrolyte) for photosynthesis, we report a design of micro-conduits in a bioinspired leaves-on-branchlet structure consisting of carbon nanotube arrays serving as branchlets and graphene petals as leaves for such electrodes. The hierarchical all-carbon micro-conduit electrodes with hollow channels exhibit high areal capacitance of 2.35 F cm^-2 (~500 F g^-1 based on active material mass), high rate capability and outstanding cyclic stability (capacitance retention of ~95% over 10,000 cycles). Furthermore, Nernst-Planck-Poisson calculations elucidate the underlying mechanism of charge transfer and storage governed by sharp graphene petal edges, and thus provides insights into their outstanding electrochemical performance.